NATIONAL UNIVERSITY OF SINGAPORE
Department of Mathematics
MA3252 Linear and Network Optimization
Tutorial 1
1. A small bank is allocating a maximum of $30,000 for personal and car loans during
the next month. The bank charges an annual interest rate of 1
Tutorial 3: Outline of Solutions
Q1. (a) For each of the following sets, decide whether it is representable as a polyhedron?
(i) The set of all (x, y ) R2 satisfying the constraints
x cos + y sin 1 [0, /2]
x 0
y 0.
(ii) The set of all x R satisfying the c
Tutorial 4: Outline of Solutions
Q1. Consider the standard form polyhedron P = cfw_x Rn |Ax = b, x 0. Suppose the
matrix A has dimensions m n and its rows are linearly independent. For each of the following
statements, state whether it is true or false. I
Tutorial 5: Outline of Solutions
Q1. Consider a minimization linear programming problem in standard form. Let x be a basic
feasible solution associated with the basis B. Prove the following:
(a) If the reduced cost of every nonbasic variable is positive,
Ch 8: Model diagnostics
I
How to check if the model fits the data well?
I
Use residual analysis!
Material:
I
I
I
Ch. 8 (we skip 8.2 for now (analysis of over-parametrized models),
but will discuss it later when discussing model selection more
formally).
Y
MTHSC 1060 Final Review Study Guide
*Note that these quick notes do not include everything that you need to know, just some of the most important
concepts. *
Chapter 1
-Composite functions from tables. Ex:
x
-1
0
1
f(x)
3
1
0
g(x)
-1
0
2
h(x)
0
-1
0
Use t
To access ROTC eBooks online
Click on http:/www.rotcebooks.net to open website (Note: Initially you
must access the ".net" site to view each eBook) Your browser must be IE9
or higher, Firefox 20 or higher or Google Chrome
Enter the following password: MOH
Transmittance
Measurement
Presented by
Dr. Richard Young
VP of Marketing & Science
Optronic Laboratories, Inc.
Optronic Laboratories, Inc.
Outline of Presentation
Types of transmittance:
Regular
Diffuse
Factors affecting measurements:
Regular Transmi
Technological progress: A burden in disguise?
Swiss Style Magazine
http:/www.swissstyle.com/technological-progress/
A preview of emerging technological trends and their societal implications
The idea of putting our planet on life support is a sobering tho
Chapter 10:
Facts & Tools
#1
a) Private cost
b) External benefit
c) External cost
d) Private benefit
e) Private cost
f) Private benefit
g) External cost
h) External benefit
#2
Yes it might reduce the undersupply of people who get flu shots. As people star
Tutorial 11: Outline of Solutions
Q1. In the following two maximum ow problems with source s = 1 and sink t = 5, determine
all s t cuts. Find the minimum s t cut in each case. Solution:
30
2
2
4
(4,0)
25
20
18
15
1
(4,6)
(0,2)
(2,0)
3
(8,0)
3
1
12
12
(6,0
Tutorial 10: Outline of Solutions
Q1. Adam and Eve are planning a drive from location A to E. The time to travel and the scenic
rating for the roads in the network are given below. All roads are one-way. Adam wants to
reach location E as fast as possible
8
O
15.999
8 is the _ and signifies the number of _ in oxygen
15.999 is both the _ and the _ of oxygen
_ mass is the weighted average mass of all isotopes of the
element
_ mass is the grams it takes of any element to equal one mole of
that element.
One mo
Decision Analysis
and Tradeoff Studies
Terry Bahill
Systems and Industrial Engineering
University of Arizona
[email protected]
, 2000-10, Bahill
This file is located in
http:/www.sie.arizona.edu/sysengr/slides/
Acknowledgement
This research was suppor
0113111311 Jllbhlllu, Wu1ua sauna 1.1.1.: nE-_-.
The liable describes a conversation hetwwn Prajapati, the creator god, and
his three species of children: gods, human beings, and demons cfw_also divine crea
tures, though adversaries of the gods). The Deus
Ch 4, part I: Moving average and autoregressive processes
I
We just finished Ch.2 and learned about stochastic processes, their
mean/autocovariance/autocorrelation functions and stationarity.
I
We now move onto discussing moving average and autoregressive
Ch. 7: Parameter estimation
I
Suppose we want to fit an ARMA model to a time series of interest:
how to estimate the parameters of the ARMA model?
I
Material part I: Ch.7.1-7.3 (7.2 is optional).
1 / 15
Overview of parameter estimation methods
I
Given an
Ch 4: Models for stationary time series
I
I
I
So far, we learned about stochastic processes and their
mean/autocovariance/autocorrelation functions, and discussed
stationary MA and AR processes.
Now, we will get into a bit more detail on the AR and MA
pro
Ch 2: Fundamental concepts
I
I
What are time series and stochastic processes?
How to describe stochastic processes?
I
I
I
Mean and autocovariance (autocorrelation) functions.
Stationarity.
Material: Ch 2, excluding the example on the random cosine wave
(p
Ch 9: Forecasting
I
Finally, lets discuss how to forecast/predict future outcomes for a
time series of interest!
I
Goal: Given Y1 , Y2 , . . . , Yt (e.g., t = n), forecast Yt+g for g > 0.
To start with:
I
I
I
I
Lets define what best predictor to use,
Lets
Ch 6: Model specification
Suppose that we are interested in the time series below;
I
I
I
I
Ch 6.1, 6.2 (PACF only, not the EACF), 6.4 (except unit root test);
the sample autocorrelation function was defined in Ch 3.6, p.49.
Well come back to the remaining
Ch. 5 (Part I): Models for non-stationary time series
I
I
In Ch. 4, we discussed stationary ARMA models. These models can
be fitted to observed time series through maximum likelihood
estimation (Ch.7) and then used for forecasting (Ch.9).
So far, for stat
Tutorial 9: Outline of Solutions
Q1. Consider the following linear programming problem.
min
s.t.
5x1 5x2 13x3
x1 + x2 + 3x3 20
12x1 + 4x2 + 10x3 90
x1 , x2 , x3 0
Let x4 and x5 denote the slack variables for the respective constraints. The optimal tableau
NATIONAL UNIVERSITY OF SINGAPORE
Department of Mathematics
MA3252 Linear and Network Optimization
Tutorial 2
1. (a) Reformulate the following problem as a linear programming problem:
max min(x1 , x2 )
s.t. |2x1 + x2 | 7
3x1 x2
0.5
1 + x1 + x2
x1 , x2 0.
Topic 1
Introduction to Linear
Programming
Example (Matrix Transpose)
Row Column
2 3 4
A=
5 8 9
x1
x2
B=
xn
Column Vector
2 5
A ' = 3 8
4 9
B ' = ( x1
x2 xn )
Row Vector
Example (Column Vectors)
Note :
For this course, an n dimensional vector x i
MA3252 LINEAR AND NETWORK
OPTIMIZATION
Topic 7: The Network Simplex Method
1 / 19
In this topic, we develop the details of the simplex method applied
to network ow problems (known as the network simplex method).
This can lead to a much faster algorithm th
MA3252 LINEAR AND NETWORK
OPTIMIZATION
Topic 6: Introduction to Network Optimization
1 / 49
Network ow problems are a special case of linear programs and
are among the most frequently solved optimization problems.
Everywhere around us networks are apparen
MA3252 LINEAR AND NETWORK
OPTIMIZATION
Topic 5: Sensitivity Analysis
1 / 27
Sensitivity (or post optimality) analysis deals with the study of
possible changes in the optimal solution as a result of making
changes in the original problem.
Why study sensiti
MA3252 LINEAR AND NETWORK
OPTIMIZATION
Topic 4: Duality Theory
1 / 44
Starting with a linear programming problem, called the primal LP,
we introduce another linear programming problem, called the dual
problem. Duality theory deals with the relation betwee